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Combinatorial Set Theory

With a Gentle Introduction to Forcing

  • Book
  • © 2017

Overview

Authors:
  1. Lorenz J. Halbeisen

    1. Department of Mathematics, ETH Zurich , Zurich, Switzerland

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  • Provides a comprehensive introduction to the sophisticated technique of forcing
  • Includes Shelah’s astonishing construction of a model in which exactly 27 Ramsey ultrafilters exist
  • Offers topics and open problems for further study
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (29 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Preliminary

    1. Front Matter

      Pages 1-1
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    2. The Setting

      • Lorenz J. Halbeisen
      Pages 3-9

    3. First-Order Logic in a Nutshell

      • Lorenz J. Halbeisen
      Pages 11-29

    4. Axioms of Set Theory

      • Lorenz J. Halbeisen
      Pages 31-83

  3. Topics in Combinatorial Set Theory

    1. Front Matter

      Pages 85-85
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    2. Overture: Ramsey’s Theorem

      • Lorenz J. Halbeisen
      Pages 87-102

    3. Cardinal Relations in ZF Only

      • Lorenz J. Halbeisen
      Pages 103-134

    4. Forms of Choice

      • Lorenz J. Halbeisen
      Pages 135-175

    5. How to Make Two Balls from One

      • Lorenz J. Halbeisen
      Pages 177-190

    6. Models of Set Theory with Atoms

      • Lorenz J. Halbeisen
      Pages 191-219

    7. Thirteen Cardinals and Their Relations

      • Lorenz J. Halbeisen
      Pages 221-243

    8. The Shattering Number Revisited

      • Lorenz J. Halbeisen
      Pages 245-257

    9. Happy Families and Their Relatives

      • Lorenz J. Halbeisen
      Pages 259-292

    10. Coda: A Dual Form of Ramsey’s Theorem

      • Lorenz J. Halbeisen
      Pages 293-315

  4. Part III

    1. Front Matter

      Pages 317-317
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    2. The Idea of Forcing

      • Lorenz J. Halbeisen
      Pages 319-321

    3. Martin’s Axiom

      • Lorenz J. Halbeisen
      Pages 323-338

    4. The Notion of Forcing

      • Lorenz J. Halbeisen
      Pages 339-368

    5. Proving Unprovability

      • Lorenz J. Halbeisen
      Pages 369-381
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Keywords

About this book

This book, now in a thoroughly revised second edition, provides a comprehensive and accessible introduction to modern set theory.

Following an overview of basic notions in combinatorics and first-order logic, the author outlines the main topics of classical set theory in the second part, including Ramsey theory and the axiom of choice. The revised edition contains new permutation models and recent results in set theory without the axiom of choice. The third part explains the sophisticated technique of forcing in great detail, now including a separate chapter on Suslin’s problem. The technique is used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In the final part, some topics of classical set theory are revisited and further developed in light of forcing, with new chapters on Sacks Forcing and Shelah’s astonishing construction of a model with finitely many Ramsey ultrafilters.


Written for graduate students in axiomatic set theory, Combinatorial Set Theory will appeal to all researchers interested in the foundations of mathematics. With extensive reference lists and historical remarks at the end of each chapter, this book is suitable for self-study.

Reviews

“Each chapter ends with Notes that often add historical information, offer further remarks on the chapter’s contents … . Halbheisen’s Combinatorial Set Theory is an excellent source for the intermediate or advanced student of set theory … Because of its wealth of material, it should also serve as an excellent resource for those designing advanced courses or searching for seminar assignments for students.” (J. M. Plotkin, zbMATH 06755662, 2018)

Authors and Affiliations

  • Department of Mathematics, ETH Zurich , Zurich, Switzerland

    Lorenz J. Halbeisen

Bibliographic Information

  • Book Title: Combinatorial Set Theory

  • Book Subtitle: With a Gentle Introduction to Forcing

  • Authors: Lorenz J. Halbeisen

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-60231-8

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing AG 2017

  • Hardcover ISBN: 978-3-319-60230-1Published: 11 January 2018

  • Softcover ISBN: 978-3-319-86812-7Published: 04 June 2019

  • eBook ISBN: 978-3-319-60231-8Published: 20 December 2017

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 2

  • Number of Pages: XVI, 594

  • Number of Illustrations: 20 b/w illustrations

  • Additional Information: Previous edition published by Springer-Verlag London, 2012

  • Topics: Mathematical Logic and Foundations, Combinatorics

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